1. Technical Field
This invention pertains to the field of wavefront sensing methods and devices, and more particularly, to a method and system for more precisely determining the location of a focal spot in a wavefront sensor, for more precisely determining the location of vertices or boundaries of lenslets in a lenslet array with respect to pixels of a wavefront sensor, for using this information to reconstruct an arbitrary wavefront with the wavefront sensor, and for more efficiently performing the calculations to perform the reconstruction.
2. Description
A light wavefront may be defined as the virtual surface delimited by all possible rays having an equal optical path length from a spatially coherent source. For example, the wavefront of light emanating from a point light source is a sphere (or a partial sphere where light from the point light source is limited to emission along a small range of angles). Meanwhile, the wavefront created by a collimating lens mounted at a location one focal length away from a point source is a plane. A wavefront may be planar, spherical, or have some arbitrary shape dictated by other elements of an optical system through which the light is transmitted or reflected.
A number of different wavefront sensors and associated methods are known. Among these are the Shack-Hartmann wavefront sensor, which will now be described in greater detail.
FIG. 1 shows a basic configuration of a Shack-Hartmann wavefront sensor 180. The Shack-Hartmann wavefront sensor 180 comprises a micro-optic lenslet array 182 that breaks an incoming beam into multiple focal spots 188 falling on an optical detector 184. Typically, the optical detector 184 comprises a pixel array, for example, a charge-coupled device (CCD) camera. The lenslets of the lenslet array 182 dissect an incoming wavefront and creates a pattern of focal spots on a charge-couple device (CCD) array. The lenslet array typically contains hundreds or thousands of lenslets, each on the size scale of a hundred microns. Most Shack-Hartmann sensors are assembled such that the CCD array is in the focal plane of the lenslet array.
A Shack-Hartmann wavefront sensor uses the fact that light travels in a straight line, to measure the wavefront of light. By sensing the positions of the focal spots 188, the propagation vector of the sampled light can be calculated for each lenslet of the lenslet array 182. The wavefront can be reconstructed from these vectors.
To better understand one or more aspects of this invention, it is worthwhile to discuss the operation of a Shack-Hartmann wavefront sensor in more detail.
In typical operation, a reference beam (e.g., a plane wave) is first imaged onto the array and the location of the focal spots is recorded. Then, the wavefront of interest is imaged onto the array, and a second set of focal spot locations is recorded. FIGS. 2A–F illustrate this process.
Usually, some optical system delivers a wavefront onto the lenslet array which samples the wavefront over the tiny regions of each lenslet. The lenslets should be much smaller than the wavefront variation, that is, the wave-front should be isoplanatic over the sampled region. When the CCD array is in the focal plane of the lenslet array, each lenslet will create a focal spot on the CCD array. The location of these focal spots is the critical measurement, for this reveals the average of the wavefront slopes across each region. That is, the shift in the focal spot is proportional to the average of the slopes of the wavefront over the region sampled by the lenslet. Software may compute the shift in each focal spot.
If the wavefront is not isoplanatic, the quality of the focal spot erodes rapidly and it becomes more difficult to determine the peak location.
However, where the isoplanatic condition is satisfied and where the focal spot shift is consistent with the small angle approximation of Fresnel, then the focal spot shift is exactly proportional to the average of the wavefront slope over the lenslet.
The incident wavefront is then reconstructed from the measurements of the average of the slopes for the hundreds or thousands of lenslets in the array.
It was stated above that the shift in the focal spot is proportional to the average of the slopes of the wavefront over the region sampled by the lenslet. This will now be shown as follows:
We begin by examining the types of wavefronts incident upon a lenslet and calculating their spatial irradiance distribution in the focal plane. We then resolve what this distribution tells us about the average value of the slopes of the sampled wavefront.
As noted above, most Shack-Hartmann sensors are assembled such that the CCD array is in the focal plane of the lenslet array. Therefore the appropriate complex amplitude distribution, Ff(u, v), of the field in the focal plane is given by the Fraunhofer diffraction pattern:
                                          F            f                    ⁡                      (                          u              ,              v                        )                          =                                            exp              ⁡                              (                                  ⅈ                  ⁢                                                                          ⁢                                      k                                          2                      ⁢                      f                                                        ⁢                                      (                                                                  u                        2                                            +                                              v                        2                                                              )                                                  )                                                    ⅈ              ⁢                                                          ⁢              λ              ⁢                                                          ⁢              f                                ⁢                      ∫                                          ∫                                  -                  ∞                                ∞                            ⁢                                                                    F                    l                                    ⁡                                      (                                          x                      ,                      y                                        )                                                  ⁢                                                                  ⁢                                  exp                  ⁡                                      (                                                                  -                        ⅈ                                            ⁢                                                                                          ⁢                                                                        2                          ⁢                                                                                                          ⁢                          π                                                                          λ                          ⁢                                                                                                          ⁢                          f                                                                    ⁢                                              (                                                  xu                          +                          yv                                                )                                                              )                                                  ⁢                                                                  ⁢                                  P                  ⁡                                      (                                          x                      ,                      y                                        )                                                  ⁢                                  ⅆ                  x                                ⁢                                  ⅆ                  y                                                                                        1        )            where the pupil function describes the lens aperture. The function is 1 inside the lens aperture and 0 outside of the aperture. In the example that follows we will build the pupil function using the rectangle function, which is defined as:
                              π          ⁡                      (            x            )                          =                  {                                                    0                                                                                  for                    ⁢                                                                                  ⁢                                                                x                                                                              >                                      1                    2                                                                                                                        1                  2                                                                                                  for                    ⁢                                                                                  ⁢                                                                x                                                                              =                                      1                    2                                                                                                      1                                                                                  for                    ⁢                                                                                  ⁢                                                                x                                                                              <                                      1                    2                                                                                                          2        )            
The field Fl(u, v) incident upon the lenslet is related to the wavefront ψ(x, y) incident upon the lenslet through:Fl(x,y)= exp(−iΨ(kx,ky))  3)
where k is the wavenumber, 2π/λ. The spatial irradiance distribution in the focal plane is related to the field amplitude by:I(x,y)=F(x,y)F*(x,y)  4)
The first step was to model the portion of the wavefront sampled by the lenslet. We begin with planar wavefronts and advance to wavefronts with more structure. Consider the first six simple cases in a Taylor series:Ψ00(x,y)=1Ψ10(x,y)=xΨ01(x,y)=yΨ20(x,y)=x2Ψ11(x,y)=xyΨ02(x,y)=y2
These six wavefronts were then analyzed using equation (2). Before presenting the results for the corresponding intensity distributions, a few functions must be defined. First is the sampling function:
                              sin          ⁢                                          ⁢          cx                =                              sin            ⁢                                                  ⁢            x                    x                                    (        5        )            
then the error function:
                              erf          ⁡                      (            x            )                          =                              2                          π                                ⁢                                    ∫              0              x                        ⁢                                          ⅇ                                  -                                      t                    2                                                              ⁢                              ⅆ                t                                                                        (        6        )            
and the imaginary error function:erƒi(x)=−ierƒ(ix)  (7)
and finally the exponential integral function:
                              ei          ⁡                      (            x            )                          =                  -                                    ∫                              -                z                            ∞                        ⁢                                                            ⅇ                                      -                                          t                      2                                                                      t                            ⁢                              ⅆ                t                                                                        (        8        )            
The intensity distributions can now be written in terms of these general functions. The first is the well-known result:
                                          I            10                    ⁡                      (                          x              ,              y                        )                          =                                            (                                                d                  2                                                  f                  ⁢                                                                          ⁢                  λ                                            )                        2                    ⁢          sin          ⁢                                          ⁢                                    c              2                        ⁡                          (                                                                    π                    ⁢                                                                                  ⁢                    d                                                        f                    ⁢                                                                                  ⁢                    λ                                                  ⁢                x                            )                                ⁢          sin          ⁢                                          ⁢                                    c              2                        ⁡                          (                                                                    π                    ⁢                                                                                  ⁢                    d                                                        f                    ⁢                                                                                  ⁢                    λ                                                  ⁢                y                            )                                                          (        9        )            
The next two results are also for planar wavefronts with tilt. They are:
                                          I            10                    ⁡                      (                          x              ,              y                        )                          =                                            (                                                d                  2                                                  f                  ⁢                                                                          ⁢                  λ                                            )                        2                    ⁢          sin          ⁢                                          ⁢                                    c              2                        ⁡                          (                                                                    π                    ⁢                                                                                  ⁢                    d                                                        f                    ⁢                                                                                  ⁢                    λ                                                  ⁢                                  (                                      x                    -                    f                                    )                                            )                                ⁢          sin          ⁢                                          ⁢                                    c              2                        ⁡                          (                                                                    π                    ⁢                                                                                  ⁢                    d                                                        f                    ⁢                                                                                  ⁢                    λ                                                  ⁢                y                            )                                                          (        10        )                        and                                                                            I            01                    ⁡                      (                          x              ,              y                        )                          =                                            (                                                d                  2                                                  f                  ⁢                                                                          ⁢                  λ                                            )                        2                    ⁢          sin          ⁢                                          ⁢                                    c              2                        ⁡                          (                                                                    π                    ⁢                                                                                  ⁢                    d                                                        f                    ⁢                                                                                  ⁢                    λ                                                  ⁢                x                            )                                ⁢          sin          ⁢                                          ⁢                                    c              2                        ⁡                          (                                                                    π                    ⁢                                                                                  ⁢                    d                                                        f                    ⁢                                                                                  ⁢                    λ                                                  ⁢                                  (                                      y                    -                    f                                    )                                            )                                                          (        11        )            
We expect the symmetry under interchange of x and yin equations (10) and (11) since space is isotropic and the lenslet has a discrete symmetry under rotations by integer multiples of 90°.
For the case of nonplanar wavefronts the results quickly become more complicated. To present these results, first define the intermediate variables:
                                                        α              =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    df                                    +                                      λ                    ⁢                                                                                  ⁢                    y                                                                    2                  ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                                                                    and                                                                                                        β              =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    df                                    -                                      λ                    ⁢                                                                                  ⁢                    y                                                                    2                  ⁢                                                                          ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                        (        12        )            
The intensity distribution can now be written as:
                                          I            20                    ⁡                      (                          x              ,              y                        )                          =                              -                          1              π                                ⁢                                    (                              d                f                            )                        2                    ⁢                      (                                          erf                ⁢                                                                  ⁢                                  i                  ⁡                                      (                                          z                      ⁢                                                                                          ⁢                      a                                        )                                                              +                              erf                ⁢                                                                  ⁢                                  i                  ⁡                                      (                                          z                      ⁢                                                                                          ⁢                      β                                        )                                                                        )                    ⁢                      (                                          erf                ⁢                                                                  ⁢                                  i                  ⁡                                      (                                          ⅈ                      ⁢                                                                                          ⁢                      z                      ⁢                                                                                          ⁢                      a                                        )                                                              +                              erf                ⁢                                                                  ⁢                                  i                  ⁡                                      (                                          ⅈ                      ⁢                                                                                          ⁢                      z                      ⁢                                                                                          ⁢                      β                                        )                                                                        )                    ⁢                                          ⁢          sin          ⁢                                          ⁢                                    c              2                        ⁡                          (                                                                    π                    ⁢                                                                                  ⁢                    d                                                        f                    ⁢                                                                                  ⁢                    λ                                                  ⁢                x                            )                                                          (        13        )            
For both I20(x,y) and I02(x,y) we see the symmetry in the solutions under interchange of x and y. The intermediate variables now become:
                                                        α              =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    df                                    +                                      λ                    ⁢                                                                                  ⁢                    x                                                                    2                  ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                                                                    and                                                                                                        β              =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    df                                    -                                      λ                    ⁢                                                                                  ⁢                    x                                                                    2                  ⁢                                                                          ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                        (        14        )            and the irradiance distribution in the focal plane is:
                                          I            02                    ⁡                      (                          x              ,              y                        )                          =                              -                          1              π                                ⁢                                    (                              d                f                            )                        2                    ⁢                      (                                          erf                ⁢                                                                  ⁢                                  i                  ⁡                                      (                                          z                      ⁢                                                                                          ⁢                      a                                        )                                                              +                              erf                ⁢                                                                  ⁢                                  i                  ⁡                                      (                                          z                      ⁢                                                                                          ⁢                      β                                        )                                                                        )                    ⁢                      (                                          erf                ⁢                                                                  ⁢                                  i                  ⁡                                      (                                          ⅈ                      ⁢                                                                                          ⁢                      z                      ⁢                                                                                          ⁢                      a                                        )                                                              +                              erf                ⁢                                                                  ⁢                                  i                  ⁡                                      (                                          ⅈ                      ⁢                                                                                          ⁢                      z                      ⁢                                                                                          ⁢                      β                                        )                                                                        )                    ⁢                                          ⁢          sin          ⁢                                          ⁢                                    c              2                        ⁡                          (                                                                    π                    ⁢                                                                                  ⁢                    d                                                        f                    ⁢                                                                                  ⁢                    λ                                                  ⁢                y                            )                                                          (        15        )            
Finally we consider the case where the input wavefront exhibits some torsion. First define the intermediate variables:
                                                                        α                1                            =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    df                                    +                                      λ                    ⁢                                                                                  ⁢                    x                                                                    2                  ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                                                                    and                                                                                                                        α                2                            =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    df                                    -                                      λ                    ⁢                                                                                  ⁢                    x                                                                    2                  ⁢                                                                          ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                        (        16        )                                                                                    β                1                            =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    df                                    +                                      λ                    ⁢                                                                                  ⁢                    y                                                                    2                  ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                                                                    and                                                                                                                        β                21                            =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    df                                    -                                      λ                    ⁢                                                                                  ⁢                    y                                                                    2                  ⁢                                                                          ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                        (        17        )            
Then the irradiance distribution can be written as:
                                          I            11                    ⁡                      (                          x              ,              y                        )                          =                                            (                              λ                                  4                  ⁢                                                                          ⁢                                      π                    2                                                              )                        2                    ⁢                      (                                          ei                ⁡                                  (                                      ⅈ                    ⁢                                                                                  ⁢                                          a                      1                                        ⁢                                          β                      1                                                        )                                            -                              ei                ⁡                                  (                                      ⅈ                    ⁢                                                                                  ⁢                                          a                      1                                        ⁢                                          β                      2                                                        )                                            -                              ei                ⁡                                  (                                                            -                      ⅈ                                        ⁢                                                                                  ⁢                                          a                      2                                        ⁢                                          β                      1                                                        )                                            +                                                          ⁢                                                          ⁢                              ei                ⁡                                  (                                                            -                      ⅈ                                        ⁢                                                                                  ⁢                                          a                      2                                        ⁢                                          β                      2                                                        )                                                      )                    ×                      (                                          ei                ⁡                                  (                                                            -                      ⅈ                                        ⁢                                                                                  ⁢                                          a                      1                                        ⁢                                          β                      12                                                        )                                            -                              ei                ⁡                                  (                                      ⅈ                    ⁢                                                                                  ⁢                                          a                      1                                        ⁢                                          β                      2                                                        )                                            -                              ei                ⁡                                  (                                      ⅈ                    ⁢                                                                                  ⁢                                          a                      2                                        ⁢                                          β                      1                                                        )                                            +                              ei                ⁡                                  (                                                            -                      ⅈ                                        ⁢                                                                                  ⁢                                          a                      2                                        ⁢                                          β                      2                                                        )                                                      )                                              (        18        )            
Clearly the solutions in equations (9)–(11) are even functions and they are separable. That is, the irradiance distributions for these three functions can be written X(x)Y(y). The next three functions are not separable as the function arguments mix the variables x and y. However, a careful analysis of the constituent functions in equations (6)–(8) reveals that the combinations in equations (13), (15) and (18) are even functions also.
Now we address the issue of how the amplitude of these input wavefronts affects the focal spot position. Consider that the wavefront has an amplitude κ. The incident wavefronts become:Ψ00(x,y)=κΨ10(x,y)=κxΨ01(x,y)=κyΨ20(x,y)=κx2Ψ11(x,y)=κxyΨ02(x,y)=κy2
These new wavefronts were again propagated through the lenslet using equation (1). As one might expect, there was no change in I00(x,y). The irradiance distributions for the odd parity terms become:
                                          I            10                    ⁡                      (                          x              ,              y                        )                          =                                            (                                                f                  ⁢                                                                          ⁢                  λ                                d                            )                        2                    ⁢          sin          ⁢                                          ⁢                                    c              2                        ⁡                          (                                                                    π                    ⁢                                                                                  ⁢                    d                                                        f                    ⁢                                                                                  ⁢                    λ                                                  ⁢                                  (                                      x                    -                                          κ                      ⁢                                                                                          ⁢                      f                                                        )                                            )                                ⁢          sin          ⁢                                          ⁢                                    c              2                        ⁡                          (                                                                    π                    ⁢                                                                                  ⁢                    d                                                        f                    ⁢                                                                                  ⁢                    λ                                                  ⁢                y                            )                                                          (        19        )                        and                                                                            I            01                    ⁡                      (                          x              ,              y                        )                          =                                            (                                                f                  ⁢                                                                          ⁢                  λ                                d                            )                        2                    ⁢          sin          ⁢                                          ⁢                                    c              2                        ⁡                          (                                                                    π                    ⁢                                                                                  ⁢                    d                                                        f                    ⁢                                                                                  ⁢                    λ                                                  ⁢                x                            )                                ⁢          sin          ⁢                                          ⁢                                    c              2                        ⁡                          (                                                                    π                    ⁢                                                                                  ⁢                    d                                                        f                    ⁢                                                                                  ⁢                    λ                                                  ⁢                                  (                                      y                    -                                          κ                      ⁢                                                                                          ⁢                      f                                                        )                                            )                                                          (        20        )            
For the higher order cases, there is a slight rewrite. For example for I20(x), the intermediate variables become:
                                                        α              =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    d                    ⁢                                                                                  ⁢                    κ                    ⁢                                                                                  ⁢                    f                                    +                                      λ                    ⁢                                                                                  ⁢                    y                                                                    2                  ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                                                                    and                                                                                                        β              =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    d                    ⁢                                                                                  ⁢                    κf                                    -                                      λ                    ⁢                                                                                  ⁢                    y                                                                    2                  ⁢                                                                          ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                        (        21        )            and the irradiance distribution in equation (13) is then multiplied by κ−1/2. For the torsion case ψ11(x, y) the intermediate variables are:
                                                                        α                1                            =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    d                    ⁢                                                                                  ⁢                    κ                    ⁢                                                                                  ⁢                    f                                    +                                      λ                    ⁢                                                                                  ⁢                    x                                                                    2                  ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                                                                    and                                                                                                                        α                2                            =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    d                    ⁢                                                                                  ⁢                    κ                    ⁢                                                                                  ⁢                    f                                    -                                      λ                    ⁢                                                                                  ⁢                    x                                                                    2                  ⁢                                                                          ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                        (        22        )                                                                                    β                1                            =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    d                    ⁢                                                                                  ⁢                    κ                    ⁢                                                                                  ⁢                    f                                    +                                      λ                    ⁢                                                                                  ⁢                    y                                                                    2                  ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                                                                    and                                                                                                                        β                2                            =                                                                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                    d                    ⁢                                                                                  ⁢                    κ                    ⁢                                                                                  ⁢                    f                                    -                                      λ                    ⁢                                                                                  ⁢                    y                                                                    2                  ⁢                                                                          ⁢                  f                  ⁢                                                                          ⁢                  λ                                                                                        (        23        )            and the irradiance distribution in equation 18 is then multiplied by κ−2.
However, the only peaks that shift position are I10(x,y) and I01(x,y). And both of these shifts are exactly in the focal plane. The question now becomes how does this compare to the average value of the wavefront slopes?
The average value of the wavefront slope is given by:
                              m          x                =                                            ∫                                                -                  d                                /                2                                            d                /                2                                      ⁢                                          ∫                                                      -                    d                                    /                  2                                                  d                  /                  2                                            ⁢                                                                    ∂                    x                                    ⁢                                      ψ                    ⁡                                          (                                              x                        ,                        y                                            )                                                                      ⁢                                  ⅆ                  x                                ⁢                                  ⅆ                  y                                                                          d            2                                              (        24        )            and similarly for my. The six input wavefronts and their average slopes m=(mx, my) are:ψ00(x,y)=κ mT=(0,0)ψ10(x,y)=κx mT=(0,0)ψ01(x,y)=κy mT=(κ,0)ψ20(x,y)=κx2 mT=(0,κ)ψ11(x,y)=κxy mT=(0,0)ψ02(x,y)=κy2 mT=(0,0)
So for these six cases, which are the most physically relevant, the average of the wavefront slope exactly describes the shift of the peak in the focal plane accounting for the distance offset of the focal length. Because of this exact relationship, it is convenient for us to define the focal spot location as being the location of the peak intensity of the focal spot.
For the purposes of this discussion, we define “isoplanatic” as the condition where the wavefront is well approximated over an area the size of the lenslet, by a plane wave.
This analysis buttresses the lore of Shack-Hartmann wavefront sensing. The shift in the focal spot position is directly proportional to the average of the wavefront slope across the lenslet. In the small angle limit inherent in the Fresnel approximation, this is an exact result for the isoplanatic cases and also true for the higher terms examined, although focal spot location is ambiguous in these cases as well as for additional higher order terms. An important issue that has not been discussed is how combinations of these terms behave. This is an important issue that is quite relevant physically.
FIGS. 3A–F and 4A–F show some exemplary incident wavefronts and output irradiance distributions in the focal plane. The lenslet is a square of size d=280μ and has a focal length f=28 mm; the wavelength λ=325 nm.
FIGS. 3A–C show the functional form of some exemplary incident wavefronts which all have a generally planar structure. FIGS. 3D–F graphically shows the corresponding wavefronts incident upon the lenslet. Note that in these examples over 95% of the light is confined to an area the size of a lenslet. The x and y axes are in microns. The vertical scale in the first column is also in microns. The vertical axis on the second column represents intensity and is in arbitrary units.
FIGS. 4A–C show a series of increasingly more complicated incident wavefronts. Clearly the focal spots are degrading rapidly where the incident wavefront has a non-planar structure. Notice the concept of a peak is ambiguous here and a center of mass computation of focal spot location could be highly problematic. Also, only a small portion of the light falls within the lenslet area. The x and y axes are in microns. The vertical scale in the first column is also in microns. The vertical axis on the second column represents intensity and is in arbitrary units.
FIGS. 3A–F and 4A–F validate the well known rule of thumb: the wavefront sensor works best in the isoplanatic limit. In other words, the lenslet size should be much smaller than the variations one is attempting to measure. Of course practical wavefront sensor design includes many other considerations too and one cannot shrink the lenslet size arbitrarily. But the message is that in the isoplanatic limit we expect to find well-defined focal spots that the center of mass can handle with higher precision.
So, to recap, a Shack-Hartmann wavefront sensor is a powerful tool for optical analysis, however the precision of the wavefront data that it produces may be less than is desired at times, and the speed and ease by which an incident wavefront may be reconstructed are also sometimes less than desirable.
First, it is difficult to precisely locate the focal spots produced by the lenslet array. Although the center of mass may be computed as an approximation of the focal spot location, more precision is desired. The basic problem with center of mass algorithms lies with deciding which pixels to include in the calculation. All cameras have a background level of electronic noise. The center of mass algorithm is especially sensitive to noise away from the brightest pixel because of weighting of the calculation by distance. To exclude this effect, a threshold is usually applied that excludes all pixels below some absolute brightness level. This threshold may be further refined after this preliminary step according to how bright the light is under a lenslet. To see the effect of this threshold, a calculation can be done using the center of mass. The sinc-squared intensity pattern can be overlaid with a discrete section of CCD pixels. The continuous intensity pattern is broken into discrete pixelated values as counts. Then the center of mass algorithm can be applied to the pixelated values. Then a calculation can be done where the sinc-squared pattern is translated into small steps across the fixed CCD. This corresponds to the situation where tilt is added in small steps to the beam as it hits the lenslet. The center of mass location can be plotted against the known tilt added to the beam. When this process is done, it will be seen that the center of mass will follow a straight line up to the point where a new pixel starts to receive enough light to be counted (or another pixel got too little light). At that point, there will be an abrupt jump in the line. The appearance will be a line with lots of little kinks in it. On the average, the input translation versus center of mass calculation will produce a line with a slope of one. So it is evident that if there are kinks in the line, the slopes of all the little straight sections of lines must be something different than the ideal slope of one. The kinks and the non-unity piecewise slopes indicates that center of mass calculations have a fundamental flaw that makes them unsuitable for being the basis of wavefront measurements if extremely high accuracies are desired.
Secondly, in general, a lenslet boundary will not line up neatly with a pixel boundary. Instead the boundary line will likely cross through pixels. For precise wavefront measurements, it is important to precisely determine the location of the vertices or boundaries of lenslets in a lenslet array with respect to pixels (e.g., CCD array) of a corresponding wavefront sensor. This knowledge is needed in order to assign locations of the boundaries of the lenslets to locations on the CCD.
Third, it is difficult to take a series of hundreds or thousands of measurements of the average of the slopes and reconstitute the incident wavefront. This process of taking the measurements and reconstituting the wavefront incident upon the lenslet array is called reconstruction and can be quite computationally intensive and time-consuming.
Accordingly, it would be desirable to provide a system and method of more accurately locating the focal spots in a wavefront sensor. It would also be desirable to provide a system and method for to precisely determine the location of the vertices or boundaries of lenslets in a lenslet array with respect to pixels (e.g., CCD array) of a corresponding wavefront sensor. Furthermore, it would be desirable to provide a system and method for reconstructing a wavefront from measurement data provided by a pixel array in a wavefront sensor. Even furthermore, it would be desirable to provide a system and method for efficiently computing polynomials for defining a wavefront detected by a wavefront sensor. And, it would be desirable to provide such a system and method which overcomes one or more disadvantages of the prior art.
The present invention comprises a system and method for sensing a wavefront.
In one aspect of the invention, a method of determining a location of a focal spot on a detector array improves upon a “pure” center of mass calculation. When a near plane wave illuminates a square lens, the focal spot has a sinc-squared shape. This knowledge can be exploited to generate a number of improvements to calculating the focal spot peak location.
Several techniques show improvement over pure center of mass methods for locating a focal spot. One technique uses the knowledge that the sinc-squared function has minimum in it before the side lobes appear. The focal spot location algorithm can be written so these minima are located, and then the pixels that are identified as being proper to include in the center of mass calculation will be those that lie within the minima. Another technique is that the shape of the sinc-square function can be fit to values that show up in each pixel. This can be done with least squares method or by “sliding” methods. Then the equation for the sinc-squared gives the location of the peak. Still another refinement is to use a different weighting than a linear weight from center in the calculation of center of mass. The effect is to reduce weight given to dimmer pixels. In particular, this weight can be matched to provide piecewise linear slopes of one, although this reduces the dynamic range of the sensor unless another method is found to eliminate the kinks.
In another aspect of the invention, a method of determining a location and size of a lenslet with respect to a detector array exploits a-priori knowledge of the spacing of the lenslets in a lenslet array. Such a-priori knowledge can be produced by direct measurement of the lenslet arrays by microphotography, by knowledge of the photolithography process, or by other means. In a particularly advantageous manufacturing process, the spacing between lenslet is extremely regular. Accordingly, when one compares algorithms that calculate focal spot locations, the better algorithm is the one that comes up with a the locations in the most regular grid. This can be done on a theoretical basis using the sinc-squared patterns as inputs, or on experimental data using a plane wave to illuminate the lenslet array. Further refinements can be obtained by adding tilt to the plane wave and doing least squares fit to determine spacings.
In a similar way, non-uniform grids can be defined if it is desired to manufacture lenslet arrays with non-uniform spacings.
In yet another aspect of the invention, a method of reconstructing a wavefront from a plurality of focal spots produced on a detector array, includes.
In still another aspect of the invention, a method of determining a set of polynomials defining a wavefront from a plurality of focal spots produced on a detector array, includes.